Modular robots have been studied an classified from different perspectives, generally focusing on the mechatronics. But the geometric attributes and constraints are the ones that determine the self-reconfiguration strategies. In two dimensions, robots can be geometrically classified by the grid in which their units are arranged and the free cells required to move a unit to an edge-adjacent or vertex-adjacent cell. Since a similar analysis does not exist in three dimensions, we present here a systematic study of the geometric aspects of three-dimensional modular robots. We find relations among the different designs but there are no general models, except from the pivoting cube one, that lead to deterministic reconfiguration plans. In general the motion capabilities of a single module are very limited and its motion constraints are not simple. A widely used method for reducing the complexity and improving the speed of reconfiguration plans is the use of meta-modules. We present a robust and compact meta-module of M-TRAN and other similar robots that is able to perform the expand/contract operations of the Telecube units, for which efficient reconfiguration is possible. Our meta-modules also perform the scrunch/relax and transfer operations of Telecube meta-modules required by the known reconfiguration algorithms. These reduction proofs make it possible to apply efficient geometric reconfiguration algorithms to this type of robots.
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